A set of premises and a conclusion are given. Use the valid argument

forms listed in Table $2\mathrm{.}3\mathrm{.}1$ to deduce the conclusion from the premises, given a reason

for each step.

a$.$ u $$ w

b$.$ u $$Table $2\mathrm{.}3\mathrm{.}1$ Valid Argument Forms

Theorem $2\mathrm{.}1\mathrm{.}1$ Logical Equivalences

Given any statement variables $p,q,$ and $r,$ a tautology $t$ and a contradiction $c,$ the following logical equivalences

hold.

Commutative Irue.

$p^{{?}^{?}}q=q^{{?}^{?}}p$

$pvvq-=qvvp$

Associative laws:

$(p^{{?}^{?}}q{)}^{{?}^{?}}r-=p^{{?}^{?}}(q^{{?}^{?}}r)$

$(pvvq)vvr-=pvv(qvvr)$

Distributive laws:

$p^{{?}^{?}}(qvvr)-=(p^{{?}^{?}}q)vv(p^{{?}^{?}}r)$

$pvv(q^{{?}^{?}}r)-=(pvvq{)}^{{?}^{?}}(pvvr)$

Identity laws:

$p^{{?}^{?}}t-=p$

$pvvc-=p$

Negation laws:

$pvvp-=t$

$p^{{?}^{?}}p-=c$

Double negative law:

$(p)-=p$

Idempotent laws:

$p^{{?}^{?}}p-=p$

$pvvp-=p$

Universal bound laws:

$pvvt-=t$

$p^{{?}^{?}}c-=c$

De Morgan's laws:

$(p^{{?}^{?}}q)pvvq$

$(pvvq)p^{{?}^{?}}q$

Absorption laws:

$pvv(p^{{?}^{?}}q)-=p$

$p^{{?}^{?}}(pvvq)-=p$

Negations of $t$ and $c$ :

$t-=c$

$c-=t$ m

c$.$ q $->$ n

d$.$ w

e$.$ m $$ r$$ n

f$.$ q